Sinusoid vibration and random vibration testing are two acceptable ways of evaluating electronics against vibration damage. Most new platforms’ vibration specs have random vibration requirements, although some sinusoid vibration requirements still exist. But many older standards only list sinusoid vibration requirements.

It would be particularly useful to relate existing sinusoid vibration test results with more current random vibration test requirements. The significance, for example, comes when an older airplane needs to upgrade its legacy systems to a new avionics suite. In this case, the newly designed avionics system is tested by random vibration under MIL-STD-810F, but the spec available on the older airplane is written in terms of sinusoidal vibration testing under MIL-STD-810A. Will the new avionics survive the vibration requirement of the old airplane without expensive retesting?

One practical example would be an electronic device designed for the new Joint Strike Fighter (JSF) and qualified to meet certain modern random vibration requirements. But if the same device is to be mounted on a B-52, the airframe’s vibration specifications are probably stated in terms of sinusoid vibration. By using the appropriate equations, the device can be qualified by analysis. Therefore, expensive double qualification testing is eliminated and total cost of the product is reduced.

This is not an academic question: COTS engineers are troubled by this problem all the time. Finding a method to relate these two vibration methods will reduce or eliminate expensive repetitive testing and qualification.

Mathematical Differences

The characterizations of sinusoid and random vibration are based on two distinctly different sets of mathematics. Sinusoid vibration testing uses a single-frequency sinusoid wave to induce vibration in the specimen by changing the vibration frequency against a time variable. The upper and lower limit of the frequency, duration of sweep, amplitude or peak acceleration of the sine wave vibration will decide the severity of the vibration. The testing equipment can be an electro-dynamics shaker or simply a mechanical shaker.

Random vibration testing uses an electro-dynamics shaker to generate multiple random frequency vibration. The duration of the testing, Power Spectra Density (PSD) level of the vibration, and the upper and lower limit of the vibration frequency will decide the severity of the vibration. Repetitive Shock (RS) shaker equipment can also generate some random vibration. The power spectral density profile is very hard to control in RS shakers.

The challenge is to relate the testing results between the two methods. Often, attempts are made to compare the peak acceleration of the sine wave to the root mean square (RMS) acceleration of random vibration. However, peak sine acceleration is the maximum acceleration at one frequency only. Random RMS is the square root of the area under a spectral density curve. These are not equivalent.

Method

The proposed method to relate random and sinusoid vibration is based on the equivalent effective damage theory. In engineering practice, it is common to equal the dynamic load to a static acceleration load to simplify the calculation, if they all cause the similar damage to the structure. In the area of aircraft structural design, all loads were calculated in terms of static load. Dynamic load was considered in terms of static load factors. A different area has a different dynamic load, and therefore has different load factors. By using equivalent static load factors and treating the dynamic load as a static load, the structural analysis is simplified.

Avionics packaging engineers had been using equivalent damage theory for many years under the direction of MIL-STD-810. The military standard had allowed the testing engineers to use the equivalent testing scenario to replace the required testing scenarios if certain conditions had been met.

In the area of avionics equipment, most components are soldered to printed circuit boards (PCBs), and the deflection of the PCB is the dominating factor to the life of the equipment. Equations are established so that the static acceleration load will cause the PCB boards to have the same deflection as the vibration load.

Another simplification method is used in the following analogy. It has been documented that the PCB boards will deflect the most at the resonance points, under the vibration load (Figure 1). In vibration analysis, any driving-frequencies, which are between two half-power points, will cause the most effective damage. The half-power point is defined as being at the driven frequency f when current transmissibility is related to maximum transmissibility Q in the following formula.

The studies described here had been performed between two half-power points of the PCB resonance points. That means before performing any kinds of conversion, the PCB resonance frequency and the maximum transmissibility has to be either calculated by finite element methods or plotted from a 1g sine sweep testing.

As soon as those system characteristics are established, the following steps can be taken to perform the conversion. First, the random vibration load is converted to a static acceleration load. The sinusoid vibration load is also converted to another static acceleration load. Then, these two converted static loads are made the same value. The sinusoid vibration load is then considered equivalent to the random vibration load. The final step would be solving the equations established above for the desired variable result (random or sinusoidal vibration).

Random to Static Load Conversion

Random vibration is a statistical concept, and the RMS acceleration value is used. Random vibration is defined by:

This equation can be transformed to the following:

This is a well-known equation and was solved by many people a long time ago. G. C. Newton, L. A. Gould and J. F. Kaiser, integrated this equation in 1957 in their book Analytical Design of Linear Feedback Controls. Later in 1975 L. Meirovitch integrated the equation with a different mathematical method. To use their result and assume a lightly damped system, the equivalent static acceleration is:

This equation has sometimes been referred as Miles’ equation. In 1954, Miles developed this equation for fatigue failure of aircraft structural components caused by engine and aerodynamic vibration. Dave Steinberg also derived the same equation with a different method, as did Thomas P. Sarafin and many other unmentioned authors. This equation will allow engineers to simplify the random vibration problems into static acceleration calculation.

Sinusoid to Static Load Conversion

Sinusoid vibration is single frequency acceleration. Two of the most popular sinusoid vibration requirements are sine sweep vibration test and sine dwell vibration test. Normally, the sine dwell vibration is constant peak sine wave acceleration at one particular frequency for a certain period of time. It is a useful tool, for example, to verify the reliability under gunfire or exposure to rotary aircraft vibration.

Sinusoid sweep vibration is involved in a constant peak sine wave acceleration whose frequency is smoothly and continuously varied. The rate of frequency is also called sweep rate and is measured in octaves.

Some sine vibration specifications require a single slow rate sweep. Some specs require multiple sweeps with a faster sweeping rate. Based on the equivalent damage theory, as soon as the duration of the sine wave at the resonance frequency is the same, there is no difference between these two methods mathematically. However, to use the equations here, a multiple sweeps standard needs to be converted to a single sweep standard first.

If sine dwell and sine sweep methods involve constant peak sine wave acceleration, the peak acceleration at a frequency f can be defined as Asin. The root mean square (RMS) value of the sine wave acceleration at the frequency f is:

After integrating the above equation, the results are:

With simplification and integration between frequency f1 and f2, the board’s root mean square corresponds to acceleration:

In constant peak acceleration sine sweep vibration, Asin is constant. By integrating the above equation (7), we will have the equivalent static acceleration G:

This is the equation of equivalent static acceleration for sinusoid vibration at the vicinity of the resonance frequency.

Random to Sinusoid Conversion

If both vibration levels generate the same deflections of the board, the equivalence can be established. Assuming that the static accelerations converted from sine and random vibration are equal, by solving the above equations the following random and sine relations are established. Converting a random vibration profile to sinusoid sweep or dwell vibration, the peak acceleration is arrived at:

The following formula will calculate the duration of the sine dwell test from random vibration requirements.

If the random vibration test requirement is converted to a sine sweep test, the sine sweep rate in octaves/second is

Sinusoid to Random Conversion

To convert sine sweep or dwell to random vibration, the following equations can be used. The total PSD level is calculated by:

The duration of random vibration trnd, can be calculated by using the total vibration cycles of sinusoid dwell divided by the resonance frequency closest to the dwell frequency:

To convert the duration of vibration trnd from sine sweep to random:

In some cases, the sine vibration consists of two parts. A sine vibration needs to sweep across the range of frequencies in addition to dwell at certain frequencies. In that case, the dwell and sweep requirements should be separately converted. The results can then be added together.

Examples and Conclusions

With some engineering simplification, it is possible to relate sinusoid vibration to random vibration. However inaccurate in science, the relationship is good enough for engineering purposes. The following example demonstrates a simple random vibration requirement that has been converted to a sinusoid vibration requirement. The second example shows the same requirement converted back from a sinusoid to a random specification.

A platform has the following random vibration specifications, as shown in Figure 2, and the component manufacturer only has a sinusoidal sweep vibration machine. The task is to convert that random spec to a new sine vibration requirement, which can then be used by the component manufacturer. The random vibration specs are: 2 hours random vibration at 0.02 g2/Hz, from 20 Hz to 2,000 Hz. The finite element analysis demonstrated that the system’s first mode natural frequency is 100 Hz with a maximum transmissibility of 5.

By using equations 9 and 11, the equivalent sinusoid vibration parameters are calculated as following:
Vibration level Asin = 2.507 g (Using Equation 9)
Rate of Sweep R = 0.000040209 Octave/
second (Using Equation 11)
Frequency Band Lower f = 20 Hz
Frequency Band Upper f = 2,000 Hz

The next example is to demonstrate—by using the equations previously derived—that the new generated sinusoidal sweep requirement can be converted back to the original random vibration specs. The same system with a first mode natural frequency of 100 Hz at Q of 5 has the following sinusoid sweep requirement: 2.5 g sine sweep from 20 Hz to 2,000 Hz, with a sweep rate of 0.000040209 Octave/second.
By using equations 12 and 13, the equivalent random vibration parameters are calculated as following:
Power Spectra Density: PSD =
0.01989479 g2/Hz (Using Equation 12)
Random Vibration Duration: trnd =
7,200 sec (Using Equation 13)
Frequency Band Lower f = 20Hz
Frequency Band Upper f = 2,000 Hz

After the sinusoid vibration is converted back to random vibration, the parameters are closely matched with the original random vibration spec.
However, there are limitations in these methods. We’ve dealt only with the two most used sinusoid vibration methods, constant peak acceleration sine sweep, as well as sine dwell tests. But if the sine sweeping profile is a complex curve, these mathematical methods will have to be modified. Additionally, if the unit has a complex system with multi major resonance points, a different kind of approach should be used.

Eldec
Lynnwood, WA.
(425) 743-8478.
[www.eldec.com].

The author wishes to acknowledge the thoughtful inputs from:
Tom Renaud of BAE Systems
Ryan Simmons of NASA Goddard Space Flight Center
Reid Ardriance of Crane Aerospace